Calculating returns
The famous job interview question (which many people miss)...
If something goes up by 50% and then decreases by 50%, how much was the overall change? It's quite annoying to realise that it's not zero after the interview and then get dinged. It's also quite annoying to find out that after losing 75% in one day in, say, online gaming shares, they will have to increase fourfold or 300% before one breaks even. What about the investor that buys £1,000 and gets a £100 profit (or a 10% return), only to lose those £100 shortly afterwards due to an annoying negative 9% return? Quite confusing!
All these problems could be solved if continuous returns were used. Continuous compounding is like receiving interest in your bank account at every millisecond. A great proposition which unfortunately does not happen often in reality. Computationally, if one receives an annual return of "R" on an initial investment of "I" compounded over "M" periods, the final investment value is:
I * [1+(R/M)] ^M
It's not difficult to imagine M becoming larger and larger and the thing becoming I * e^R. That is actually the definition of the constant "e".
In the world of continuous compounding, if something goes up 50% and then down 50%, we are back to square zero. My online gaming shares would have gone down by 139% (which is ln(250)-ln(1,000)) and they need to go up 139% for me to break even. The £100 profit on the £1,000 investment would mean a return of 9.5% and subsequent loss would represent a return of minus 9.5%. Furthermore, multiple year returns become basic calculations. Five consecutive years of 10% returns are no longer equal to a confusing 61% return, but rather a straightforward 50%.
Of course, there are some annoying aspects as well. A doubling of one's initial investment is no longer a nice 100% round return, but a cryptic 69.3147% continuous return. And, if one loses all the initial investment, there is no other way to put it than an infinitely negative return, which, well, might be a more accurate representation of such tragedy.
If something goes up by 50% and then decreases by 50%, how much was the overall change? It's quite annoying to realise that it's not zero after the interview and then get dinged. It's also quite annoying to find out that after losing 75% in one day in, say, online gaming shares, they will have to increase fourfold or 300% before one breaks even. What about the investor that buys £1,000 and gets a £100 profit (or a 10% return), only to lose those £100 shortly afterwards due to an annoying negative 9% return? Quite confusing!
All these problems could be solved if continuous returns were used. Continuous compounding is like receiving interest in your bank account at every millisecond. A great proposition which unfortunately does not happen often in reality. Computationally, if one receives an annual return of "R" on an initial investment of "I" compounded over "M" periods, the final investment value is:
I * [1+(R/M)] ^M
It's not difficult to imagine M becoming larger and larger and the thing becoming I * e^R. That is actually the definition of the constant "e".
In the world of continuous compounding, if something goes up 50% and then down 50%, we are back to square zero. My online gaming shares would have gone down by 139% (which is ln(250)-ln(1,000)) and they need to go up 139% for me to break even. The £100 profit on the £1,000 investment would mean a return of 9.5% and subsequent loss would represent a return of minus 9.5%. Furthermore, multiple year returns become basic calculations. Five consecutive years of 10% returns are no longer equal to a confusing 61% return, but rather a straightforward 50%.
Of course, there are some annoying aspects as well. A doubling of one's initial investment is no longer a nice 100% round return, but a cryptic 69.3147% continuous return. And, if one loses all the initial investment, there is no other way to put it than an infinitely negative return, which, well, might be a more accurate representation of such tragedy.
posted by DS at 4:00 PM
<< Home